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Math Help - Inverse function

  1. #1
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    Inverse function

    Find the inverse of the function:

    f(t) = (10^t - 10^(-t))/(10^t + 10 ^ (-t))

    Thanks,
    Anshu
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  2. #2
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    Quote Originally Posted by Curious_eager View Post
    Find the inverse of the function:

    f(t) = (10^t - 10^(-t))/(10^t + 10 ^ (-t))

    Thanks,
    Anshu
    f(t)=y=\dfrac{10^t-\dfrac1{10^t}}{10^t+\dfrac1{10^t}} = \dfrac{10^{2t}-1}{10^{2t}+1}

    Solve the equation for t:

    y= \dfrac{10^{2t}-1}{10^{2t}+1} Substitute a = 10^{2t}

    y= \dfrac{a-1}{a+1}~\implies~a=\dfrac{y+1}{y-1}

    Now re-substitute:

    10^{2t}=\dfrac{y+1}{y-1}~\implies~2t=\log\left(\dfrac{y+1}{y-1}\right)~\implies~\boxed{t=\dfrac12\cdot \log\left(\dfrac{y+1}{y-1}\right)}
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